**Problem : **

Give four different definitions of the chemical potential *μ*, as
derivatives of the different energies we have defined.

*μ* = = = =

**Problem : **

Give two definitions of the entropy *σ* in terms of derivatives of
the different energies we have defined.

*σ* = - = -

**Problem : **

Using the definition of temperature that uses the enthalpy, give an
expression for the temperature in terms of *U*, *σ*, *p*, and *V*,
following the method used to derive an expression for the pressure
above.

We know that *τ* = , and
that *H* = *U* + *pV*. We can differentiate the second equation with respect to
*σ*, holding *p* and *N* constant, and then set equal to *τ* to
obtain:

**Problem : **

Derive the Maxwell relation that relates a derivative of *μ* with a
derivative of *σ*.

We use *G* because *μ* and *σ* are free in its differential
identity. We can write = *μ* and = - *σ*.
Taking the partial derivative of the first with respect to *τ*, holding
*N* constant, and taking the partial derivative of the second with respect to
*N*, holding *τ* constant, and setting the two equal, we obtain:

= -

**Problem : **

Derive the Maxwell Relation that relates a derivative of *τ* with a
derivative of *V*.

We need *V* and *τ* to be free in the energy, so let us choose the
enthalpy *H*. Then we can write *τ* = and *V* = .
Taking the partial derivative of the first with respect to *p*, holding
*σ* constant, and taking the partial derivative of the second with
respect to *σ*, holding *p* constant, and setting them equal,
yields:

=

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